Structural Mechanics

Problem Statement:

Design a cantilever beam with a circular cross-section to support a load, limiting the maximum bending stress.

• Span (L): 3.2 meters
• Maximum Allowable Bending Stress (σ_max): 15 N/mm²

Assumptions:

• The load is applied at the free end of the cantilever beam.
• The material is homogeneous and isotropic.
• The beam is perfectly straight.
• Shear stress is not a primary design factor in this case. We are focusing on bending stress.

Design Steps:

1. Determine the Maximum Bending Moment (M):

   First, we need to assume a load (W) at the free end of the beam. Let's assume a load for now and iterate if the resulting diameter is impractical. Assume W = 100 N.

   For a cantilever beam with a point load at the free end, the maximum bending moment occurs at the fixed end and is calculated as:

   M = W * L

   M = 100 N * 3200 mm = 320,000 N·mm

2. Calculate the Required Section Modulus (Z):

   The bending stress (σ) is related to the bending moment (M) and section modulus (Z) by the following formula:

   σ = M / Z

   Therefore, the required section modulus is:

   Z = M / σ_max

   Z = 320,000 N·mm / 15 N/mm² = 21,333.33 mm³

3. Determine the Diameter (d) of the Circular Cross-Section:

   For a circular cross-section, the section modulus (Z) is given by:

   Z = (π * d³) / 32

   Where 'd' is the diameter of the circle.

   Solving for 'd':

   d³ = (32 * Z) / π

   d³ = (32 * 21,333.33 mm³) / π ≈ 217,146.98 mm³

   d = ∛217,146.98 mm³ ≈ 60.07 mm

4. Select a Suitable Diameter:

   Choose a standard or readily available diameter slightly larger than the calculated value. For example, select d = 65 mm. This provides a small safety margin.

5. Verify the Bending Stress:

   Recalculate the section modulus (Z) with the chosen diameter:

   Z = (π * (65 mm)³) / 32 ≈ 27,150.75 mm³

   Calculate the actual bending stress:

   σ = M / Z = 320,000 N·mm / 27,150.75 mm³ ≈ 11.79 N/mm²

   Since 11.79 N/mm² < 15 N/mm², the design is acceptable for the assumed load of 100N.

Iteration (Important!):

The above calculation was based on an assumed load of 100 N. This is unlikely to be the *actual* load. The *actual* load the beam will experience needs to be determined through a different method.

To complete the design:

1. Determine the actual load (W) that the cantilever beam needs to support based on the application.
2. Repeat steps 1-5 using the *actual* load (W).

Without knowing the real load, it is impossible to complete the beam design. This process shows how to select the correct diameter when that load is known.

Important Considerations:

• Material Selection: The material's yield strength must be significantly higher than the calculated bending stress, including a factor of safety. Common materials include steel, aluminum, or composites. The elastic modulus (Young's Modulus) is also important for deflection calculations (not performed here).
• Deflection: Calculate the maximum deflection of the cantilever beam to ensure it meets the application's requirements. The deflection should be within acceptable limits.
• Shear Stress: While bending stress is the primary concern here, check shear stress, especially near the fixed end. For circular cross-sections, shear stress is often less critical than bending stress, but should still be checked.
• Buckling: If the beam is very long relative to its diameter, consider buckling. This is less likely to be an issue for typical cantilever beam proportions but should be checked for slender beams.
• Manufacturing Tolerances: Account for manufacturing tolerances in the diameter.

1. Problem Definition:

• Span (L): 3.2 meters (3200 mm)
• Maximum Allowable Bending Stress (σ_max): 50 N/mm²
• Beam Type: Cantilever with circular cross-section

2. Material Selection:

We will initially consider structural steel (e.g., ASTM A36) due to its common usage and availability. However, the final material selection depends on factors like cost, weldability, and desired lifespan. For initial calculations, assume:

• Yield Strength (σ_y): 250 N/mm² (for A36 steel - this is just for reference and not used in the bending stress calculation directly but informs the factor of safety) Engineering ToolBox - ASTM A36 Steel

3. Load Estimation:

This is the most critical and least defined part of the problem. We must estimate the maximum force (F) that the bending press will apply at the end of the cantilever beam. Without this, we cannot complete the design.

Assume for this example: The bending press will exert a maximum force of 10,000 N (1 metric ton) at the end of the beam.

4. Bending Moment Calculation:

For a cantilever beam with a point load (F) at the free end, the maximum bending moment (M) occurs at the fixed end and is given by:

M = F * L

M = 10,000 N * 3200 mm = 32,000,000 N·mm

5. Section Modulus Calculation:

The bending stress (σ) is related to the bending moment (M) and section modulus (Z) by:

σ = M / Z

Therefore, Z = M / σ

Z = 32,000,000 N·mm / 50 N/mm² = 640,000 mm³

6. Diameter Calculation:

For a circular cross-section, the section modulus (Z) is given by:

Z = (π * d³) / 32

Where 'd' is the diameter of the circular cross-section.

Therefore, d³ = (32 * Z) / π

d³ = (32 * 640,000 mm³) / π ≈ 6,517,946 mm³

d = ∛6,517,946 mm³ ≈ 187.1 mm

Therefore, we will choose a diameter of 190 mm as a starting point for our design.

7. Deflection Check:

It's essential to verify that the deflection is within acceptable limits. The maximum deflection (δ) for a cantilever beam with a point load at the free end is:

δ = (F * L³) / (3 * E * I)

Where:

• E is the modulus of elasticity of the material (e.g., for steel, E ≈ 200,000 N/mm²)
• I is the second moment of area (moment of inertia) of the circular cross-section. For a circle, I = (π * d⁴) / 64

Calculate I: I = (π * (190 mm)⁴) / 64 ≈ 63.9 x 10⁶ mm⁴

Calculate Deflection: δ = (10,000 N * (3200 mm)³) / (3 * 200,000 N/mm² * 63.9 x 10⁶ mm⁴) ≈ 8.52 mm

This deflection should be assessed for suitability in the application. If the deflection is too high, consider increasing the diameter further.

8. Factor of Safety:

The design should incorporate an appropriate factor of safety. A common factor of safety for structural steel applications is between 1.5 and 3. Since we are limiting bending stress to 50 N/mm², which is much smaller than the yield strength, the factor of safety is intrinsically high. However, buckling needs to be considered.

9. Buckling Check:

Cantilever beams under bending can be susceptible to lateral-torsional buckling, especially with long spans. This is particularly true for beams with cross-sections that are not very rigid in the lateral direction. A circular cross-section is excellent in this regard, but it should still be checked, particularly if the force estimate turns out to be significantly higher.

A detailed buckling analysis requires more information about the specific application and support conditions. Simplified buckling checks can be performed using Euler's formula with appropriate effective length factors. This is a critical step that requires specialist engineering knowledge, and must be investigated during detailed design.

10. Assembly and Support:

The fixed end of the cantilever beam must be rigidly supported. This usually involves welding or bolting the beam to a robust frame or structure. The support structure must be capable of withstanding the reaction forces and moments from the beam. The design of the support is as important as the beam itself.

• Welding: If welding is used, ensure the weld is strong enough to withstand the bending moment and shear force at the fixed end. Use appropriate welding procedures and qualified welders.
• Bolting: If bolting is used, calculate the required number and size of bolts based on the reaction forces and moments. Use high-strength bolts and ensure proper pre-tensioning.

11. Further Considerations:

• Manufacturing Tolerances: Account for manufacturing tolerances in the diameter and length of the beam.
• Surface Finish: Consider the required surface finish for the application.
• Corrosion Protection: Implement appropriate corrosion protection measures, such as painting or galvanizing, if the beam will be exposed to corrosive environments.
• Dynamic Loading: If the bending press will be subjected to dynamic loads, perform a fatigue analysis to ensure the beam can withstand repeated loading cycles.
• Finite Element Analysis (FEA): For a more detailed analysis, especially concerning buckling and stress concentrations at the support, consider using FEA software.

Summary

• Based on the assumptions above (a 10,000 N load), a 190 mm diameter circular steel beam is a reasonable starting point.
• Buckling must be investigated by a qualified engineer.
• The deflection check is important and must be assessed in the application.
• The fixed end support design is critical and must be robust.

Shear Force (SF):

• Shear force at a section is the algebraic sum of all the vertical forces acting either to the left or to the right of the section.
• It represents the internal force within the beam that resists the tendency of one part of the beam to slide or shear relative to the other part.
• A shear force diagram plots the variation of this shear force along the length of the beam.

Bending Moment (BM):

• Bending moment at a section is the algebraic sum of the moments of all the forces acting either to the left or to the right of the section about that section.
• It represents the internal moment within the beam that resists the bending caused by the applied loads.
• A bending moment diagram plots the variation of this bending moment along the length of the beam.

Purpose and Use:

• Determining Internal Forces: SFBMDs help determine the magnitude and location of maximum shear force and bending moment, which are critical for structural design.
• Structural Design: Engineers use these diagrams to select appropriate beam sizes and materials to ensure the structure can withstand the internal forces and moments without failure.
• Identifying Critical Points: They help identify points of maximum stress and deflection, which are important for ensuring the structural integrity and serviceability of the beam.
• Understanding Beam Behavior: SFBMDs provide a visual representation of how the beam responds to different loading conditions, aiding in understanding its overall structural behavior.

How to Create:

• Calculate Support Reactions: Determine the reactions at the supports of the beam.
• Establish Sections: Divide the beam into sections based on the locations of applied loads and supports.
• Calculate Shear Force and Bending Moment Equations: For each section, derive equations for shear force and bending moment as a function of the distance along the beam.
• Plot the Diagrams: Plot the shear force and bending moment equations on a graph, with the beam's length on the x-axis and the shear force/bending moment on the y-axis.

Sign Conventions:

• Shear Force: A common convention is to consider upward forces to the left of the section as positive and downward forces to the right of the section as positive.
• Bending Moment: A common convention is to consider moments that cause compression in the upper fibers of the beam (sagging) as positive and moments that cause tension in the upper fibers (hogging) as negative.